Improvement on the axial force accuracy of the ANCF Euler-Bernoulli beam element with the second-order approximate function of the centerline and precise constraint equations

The conventional Euler-Bernoulli beam element of the absolute nodal coordinate formulation (ANCF) yields incorrect axial force due to the existence of the first-order approximate function of the centerline and over-constraint equations. This work aims to improve the axial force accuracy of the element by using the second-order centerline approximate function and precise constraint equations. The second-order function includes the quintic Hermite shape function and the vector of nodal coordinates containing zeroth to second-order derivatives of position. The elastic force vector of the element and its tangent stiffness were then formulated based on the decoupled elastic line method. The exact kinematic requirements of three types of constraint points were fully clarified. On this basis, the corresponding concise constraint equations, constraint Jacobian and Hessian matrices were rigorously derived in the context of ANCF, where G2 continuity was demonstrated to be satisfied at the node acted by an inclined concentrated force, the relative rotation constraint of the rigid joint was described using the dot product of the gradients on both sides of the joint, and the fixing gradient direction at the partially-clamped end was expressed by using the zero cross product of the end gradients in the initial and deformed configurations. Large deformation problems of typical beams under fixed and follower loads were solved using five types of elements, where the results of the ABAQUS B22 elements were taken as a reference for validation. It is found that when using the first- and second-order beam elements having over-constraint equations of the three types of constraint points, obvious axial force error occurs near the corresponding constraint points. In comparison, a few second-order elements containing the derived constraint equations output correct axial force consistent with ABAQUS. However, a large number of the first-order elements containing the correct constraint equations still yield inaccurate axial force in certain beam regions.